Name: estimating area with the midpoint rule
Uploaded: 2026-01-12T12:24:26.000Z
Duration: 1 min 20 s
Channel: JoVE

La règle du point médian aide à estimer l’aire sous une courbe en divisant l’intervalle en sous-intervalles égaux.Le point médian de chaque sous-intervalle est calculé comme la moyenne de ses deux extrémités. La valeur de la fonction à ces points médians donne la hauteur des rectangles dessinés au-dessus. Cette approche donne souvent une estimation plus précise que les méthodes utilisant les extrémités gauche ou droite, car elle réduit les erreurs causées par la sous-estimation ou la surestimation de la surface.Considérez une région au bord d’une rivière. Elle est délimitée par deux lignes verticales sur les côtés, une ligne horizontale en dessous, et une rive courbe au-dessus.Pour estimer sa surface, la région est divisée en plusieurs rectangles de largeur égale.La hauteur de chaque rectangle est déterminée à l’aide de la fonction au milieu de sa base.L’aire de chaque rectangle est déterminée en multipliant sa hauteur par la largeur du sous-intervalle. Ces aires sont additionnées pour approximer la surface totale sous la courbe, donnant la formule de la règle du point médian.À mesure que le nombre de rectangles augmente et qu’ils se rétrécissent, l’estimation s’améliore et approche la surface réelle.

méthode du point milieu

L&#x2019;approximation des aires sous des courbes constitue un probl&#xE8;me fr&#xE9;quent en math&#xE9;matiques appliqu&#xE9;es, en particulier lorsque le calcul exact s&#x2019;av&#xE8;re difficile ou impraticable. La m&#xE9;thode du point milieu est une m&#xE9;thode num&#xE9;rique efficace &#xE0; cet effet : elle permet d&#x2019;estimer l&#x2019;aire sous une courbe au moyen d&#x2019;approximations rectangulaires sur un intervalle donn&#xE9;.
Description de la m&#xE9;thode du point milieu
La m&#xE9;thode du point milieu commence par diviser l&#x2019;intervalle donn&#xE9; en un certain nombre de sous-intervalles de m&#xEA;me longueur. Pour chaque sous-intervalle, le point milieu est d&#xE9;termin&#xE9; en calculant la moyenne des valeurs &#xE0; ses deux bornes. La valeur de la fonction en ce point milieu est ensuite utilis&#xE9;e pour d&#xE9;terminer la hauteur d&#x2019;un rectangle construit sur ce sous-intervalle. Comme le point milieu se situe g&#xE9;n&#xE9;ralement plus pr&#xE8;s du comportement moyen de la fonction sur cet intervalle, cette m&#xE9;thode tend &#xE0; r&#xE9;duire les erreurs de surestimation ou de sous-estimation qui peuvent survenir lorsque l&#x2019;on utilise les bornes gauche ou droite.
Interpr&#xE9;tation g&#xE9;om&#xE9;trique
Un exemple concret de la m&#xE9;thode du point milieu consiste &#xE0; estimer l&#x2019;aire d&#x2019;une r&#xE9;gion voisine d&#x2019;une rivi&#xE8;re. Cette r&#xE9;gion est d&#xE9;limit&#xE9;e &#xE0; gauche et &#xE0; droite par des droites verticales, en bas par une droite horizontale et en haut par une berge courbe. La fronti&#xE8;re sup&#xE9;rieure &#xE9;tant irr&#xE9;guli&#xE8;re, son aire ne peut pas &#xEA;tre calcul&#xE9;e ais&#xE9;ment &#xE0; l&#x2019;aide de formules g&#xE9;om&#xE9;triques simples.
Processus d&#x2019;approximation de l&#x2019;aire
Pour approcher l&#x2019;aire de cette r&#xE9;gion, elle est divis&#xE9;e en plusieurs rectangles de largeur &#xE9;gale. La hauteur de chaque rectangle est d&#xE9;termin&#xE9;e en &#xE9;valuant la courbe fronti&#xE8;re au point milieu de la base du rectangle. L&#x2019;aire de chaque rectangle est calcul&#xE9;e en multipliant sa hauteur par sa largeur. La somme des aires de tous les rectangles fournit une estimation de l&#x2019;aire totale sous la courbe, ce qui constitue le principe fondamental de la m&#xE9;thode du point milieu.
Pr&#xE9;cision de la m&#xE9;thode
&#xC0; mesure que le nombre de rectangles augmente et que leur largeur diminue, l&#x2019;approximation par rectangles suit de plus en plus fid&#xE8;lement la forme de la courbe. Par cons&#xE9;quent, l&#x2019;aire estim&#xE9;e devient plus pr&#xE9;cise et se rapproche de l&#x2019;aire r&#xE9;elle de la r&#xE9;gion.

Méthode du point milieu

L&#x2019;approximation des aires sous des courbes constitue un probl&#xE8;me fr&#xE9;quent en math&#xE9;matiques appliqu&#xE9;es, en particulier lorsque ...

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Midpoint Rule

estimating area with the midpoint rule

The Midpoint Rule estimates the area under a curve by using rectangles. It is a numerical method that helps when an exact area is hard to find or not practical to calculate. By using rectangle heights from midpoints, it gives a useful approximation over a chosen interval.

To use the Midpoint Rule, the interval is split into equal subintervals. The midpoint of each subinterval is found by averaging its two endpoints. The function value at that midpoint gives the height of a rectangle over that subinterval.

This midpoint choice often matches the average behavior of the curve better than using the left or right endpoint. That can reduce the error that may happen with overestimation or underestimation. In this way, the method gives a closer area estimate than simple endpoint-based rectangles in many cases.

The idea can be shown with a region next to a river. The region is bounded on the left and right by vertical lines, below by a horizontal line, and above by a curved riverbank. Since the upper edge is irregular, its area is not easy to find with simple geometry.

The area is approximated by dividing the region into several rectangles of equal width. Each rectangle’s area is found by multiplying its width by its midpoint height. Adding the areas of all the rectangles gives the estimated total area under the curve.

The estimate improves as the number of rectangles increases and their widths get smaller. The rectangles then follow the curve more closely. As a result, the approximation approaches the true area of the region.

Estimating Area with the Midpoint Rule

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key terms and concepts

Learn how the Midpoint Rule estimates area under a curve with equal-width rectangles, midpoint heights, and better accuracy as intervals get smaller.

Estimating Area with the Midpoint Rule | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Estimating Area with the Midpoint Rule

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